Anisotropic parabolic equations with variable nonlinearity

By Stanislav N. Antontsev, S. Shmarev. 2009

209 citations recorded by the source corpus.

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/3f5ac1fee4203ef587026afabc77a4b7027d6e6b

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Calderón–Zygmund estimates for parabolic p(x,t)-Laplacian systems (2014, background): “The existence of such weak solutions is ensured by a result of Antontsev and Shmarev [7], [8].”
Hölder estimates for parabolic p(x, t)-Laplacian systems (2012, result): “The existence of such weak (energy) solutions is ensured by a result of Antontsev and Shmarev [4,5].”
Existence of solutions to an initial Dirichlet problem of evolutional p(x)-Laplace equations☆ (2012, methodology): “Unlike [3] , we will, in this paper, adopt a method of difference in time.”
Nonlocal $p$-Laplace equations depending on the $L^p$ norm of the gradient (2014, background): “During the last decade, many mathematicians have been studying problems associated with the p-Laplace operator, which appears in a variety of physical fields (see for instance [1], [2]).”
Monotone operator theory for unsteady problems in variable exponent spaces (2012, methodology): “To our knowledge the first results which allow the treatment of parabolic equations with variable exponents by using classical tools such as Minty’s trick for monotone operators go back to Antontsev and Shmarev [18,19].”
Wave Equation with p(x,t) - Laplacian and Damping Term: Existence and Blow-up (2011, result): “By analogy with [11] we prove the following propositions.”